A note on optimal designs for a two-part model

D- and DA-optimal designs are investigated for a model where the response is a mixture of zero and a lognormal random variable, and can be modelled using a logit model combined with a linear regression on the logarithm of the lognormal part. The DA-optimal design for estimating the logit model parameters is shown to coincide with the D-optimal design for the usual logit model. The DA-optimal design for estimating the linear model parameters is derived analytically. Among two-point designs, an analytic solution is given for the design that maximizes the D-optimality.